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September 17th, 2013, 11:46 PM   #1
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The empty set

If P(A) is the power set of A=[a,b] and we no that B is a non-empty subset of A does B might be the empty set ? On one hand I just said the B is non-empty, on the other P(A) includes the empty set, so being the empty set B still includes an actual element of P(A). I am confused.
Thank you.
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September 18th, 2013, 02:07 AM   #2
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Re: The empty set

First, the empty set is a subset of any set. But, this doesn't mean that it contains any element of the set.

Here's an example that may help:

Suppose you have a choice of a Coke or a Sprite. You actually have four options:

You could drink the Coke.
You could drink the Sprite.
You could be greedy and drink both!
Or, you could drink neither.

This last option (not to choose anything that is available) is logically the same as the empty set being a subset of any set.
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September 19th, 2013, 07:32 AM   #3
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Re: The empty set

Quote:
Originally Posted by barokas
If P(A) is the power set of A=[a,b] and we no that B is a non-empty subset of A does B might be the empty set ? On one hand I just said the B is non-empty, on the other P(A) includes the empty set, so being the empty set B still includes an actual element of P(A). I am confused.
Thank you.
Your post is confused! "If P(A) is the power set of A=[a,b] and we know that B is a non-empty subset of A does B might be the empty set ?" No, if we "know B is non-empty subset of A" then B is not the empty set!

Had you just said "B is a subset of A" or "B is a member of P(A)" then B might well be empty. Yes, P(A) includes the empty set but that has nothing to do with "includes an actual element of A". (Another confusion- "actual elements of P(A)", which is what you wrote, are subsets of A. "Subsets of A" do NOT generally contain other subsets of A.)

You may be confused about the definition of "subset of A". Saying that "B is a subset of A" does NOT necessarily mean B "includes an actual element of A". B is a subset of A if B does NOT include any element that is NOT in A. Since the empty set does NOT include any elements, it cannot include any that are "not in A" and so is a subset of any set, A.

P(A) is the set of all subsets of A, NOT just "all non-empty subsets of A" and always includes the empty set.
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September 22nd, 2013, 09:32 PM   #4
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Re: The empty set

Thanks, my question was indeed confusing as i wanted to ask if b may contain the emty set i learned that the answer isnpositive. Thanks again
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September 25th, 2013, 04:47 PM   #5
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Re: The empty set

Quote:
Originally Posted by barokas
Thanks, my question was indeed confusing as i wanted to ask if b may contain the emty set i learned that the answer isnpositive. Thanks again
If B is the power set of A, then B not just can but invariably does contain the empty set. You never defined B.
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